Problem 64 - Algebra 2

Question

ACT/SAT The measures of the angles of a triangle form an arithmetic sequence. If the measure of the smallest angle is \(36^{\circ},\) what is the measure of the largest angle? $$ \begin{array}{llllll}{\mathbf{A}} & {75^{\circ}} & {\mathbf{B}} & {84^{\circ}} & {\mathbf{C}} & {90^{\circ}} & {\mathbf{D}} & {97^{\circ}}\end{array} $$

Step-by-Step Solution

Answer

The largest angle is \(84^{\circ}\) which corresponds to option B.

Step 1: Understand the Arithmetic Sequence

In an arithmetic sequence, each term after the first is found by adding a constant difference to the previous term. In this case, the angles of a triangle form an arithmetic sequence with the smallest angle given as \(36^{\circ}\). Let the common difference be \(d\). So, the three angles are \(36^{\circ}\), \(36^{\circ} + d\), and \(36^{\circ} + 2d\).

Step 2: Use the Triangle Angle Sum Property

The sum of the angles in a triangle is always \(180^{\circ}\). Therefore, we can write the equation for the sum of the angles in this triangle as follows: \[36^{\circ} + (36^{\circ} + d) + (36^{\circ} + 2d) = 180^{\circ}\].

Step 3: Simplify and Solve for d

Simplify the left side to combine like terms: \[108^{\circ} + 3d = 180^{\circ}\]. Solve for \(d\) by subtracting \(108^{\circ}\) from both sides: \[3d = 72^{\circ}\]. Divide both sides by 3 to find \(d\): \[d = 24^{\circ}\].

Step 4: Calculate the Largest Angle

The largest angle is the third angle in the sequence, which is \(36^{\circ} + 2d\). Substitute \(d = 24^{\circ}\) into the expression: \[36^{\circ} + 2 \times 24^{\circ} = 36^{\circ} + 48^{\circ} = 84^{\circ}\].

Key Concepts

Angle Measures

Understanding angle measures is crucial, especially in the context of triangles. Angles are typically measured in degrees, and in a triangle, there are three angles to consider. These angle measures can often be organized into specific mathematical sequences. For instance, in an **arithmetic sequence**, each angle measure increases by a constant difference. This sequential arrangement allows for systematic calculations,
making it easier to predict unknown angles based on known values. For example, if we know the smallest angle, we can add the common difference to find the next angle measure. Adding this difference again leads us to the largest angle. Arithmetic sequences provide a straightforward method to organize and compute angle measures efficiently. In various exercises and problems, recognizing how these measures relate can offer significant insights.

Triangle Properties

Triangles are intriguing shapes defined by specific properties. The essential property to bear in mind is that every triangle has three edge lengths and three angles. These basic elements dictate what type of triangle you have.
For example, if all angles are equal, the triangle is equilateral. If only two angles are equal, it's isosceles. If all angles have different measures, it's a scalene triangle.

• **Interior Angles**: The sum of these angles decides if specific angle constraints are met.
• **Sides**: The length of each side connects directly to the triangle's angles.
• **Congruence and Similarity**: Angles help determine if triangles are either congruent (same shape and size) or similar (same shape but different sizes).

Understanding these properties aids in identifying relationships within the triangle, leading to more accurate problem-solving.

Sum of Angles in a Triangle

It's vital to know that the sum of all interior angles in a triangle is always exactly 180 degrees. This is known as the **Triangle Angle Sum Property**. It's a fundamental rule in geometry that aids in solving various problems.
When working with exercises involving triangles, applying this rule helps validate calculations or solve for unknown angles. In an arithmetic sequence, the angles are systematically structured. Therefore, understanding the sum property allows you to form equations that incorporate all angles of a triangle:

• Start by recognizing that each part of the sequence contributes to the total 180 degrees.
• Combine known angles and differences to find unknown angles efficiently, as seen in our example problem where arithmetic sequencing determines specific angle values.
• Using the sum effectively checks and balances the relations between angles.

Grasping the sum of angles offers a solid foundation in triangle-related problems and proofs, enabling you to tackle more complex geometric questions with confidence.